Layer cake representation

In mathematics, the layer cake representation of a non-negative, real-valued measurable function

defined on a measure space

is the formula for all

denotes the indicator function of a subset

denotes the super-level set The layer cake representation follows easily from observing that and then using the formula The layer cake representation takes its name from the representation of the value

as the sum of contributions from the "layers"

contribute to the integral, while values

It is a generalization of Cavalieri's principle and is also known under this name.[1]: cor.

2.2.34 An important consequence of the layer cake representation is the identity

μ ( x ) =

μ ( { x ∈

which follows from it by applying the Fubini-Tonelli theorem.

An important application is that

can be written as follows

μ ( x ) = p

μ ( { x ∈

which follows immediately from the change of variables

in the layer cake representation of

This representation can be used to prove Markov's inequality and Chebyshev's inequality.